# Lesson Worksheet: Linear Transformation Composition Mathematics

In this worksheet, we will practice finding the matrix of two or more consecutive linear transformations.

Q1:

Let the matrix represent rotation in the plane through an angle of and let the matrix represent reflection in the -axis.

What is the matrix ?

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What is the matrix ?

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What is the matrix ?

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Q2:

Suppose that the matrix represents rotation about the origin through an angle of (measuring between and ) and the matrix represents reflection in the .

Find .

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Note that is a reflection in a line through the origin. Let this line of reflection have equation . By considering the image of the vector , determine the measure of the angle between the and the line of reflection.

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What, therefore, is the slope of the line of reflection?

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If lies in the direction of the line of reflection, what is ?

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By solving the equation obtained in the previous part, find another expression for the slope of the line of reflection.

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Q3:

Suppose and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?

• AA reflection in the line through the origin at a inclination
• BA reflection in the line through the origin at a inclination
• CA reflection in the line through the origin at a inclination
• DA reflection in the line through the origin at a inclination
• EA reflection in the line through the origin at a inclination

Q4:

Suppose and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?

• AA reflection in the line through the origin at a inclination
• BA reflection in the line through the origin at a inclination
• CA reflection in the line through the origin at a inclination
• DA reflection in the line through the origin at a inclination
• EA reflection in the line through the origin at a inclination

Q5:

Describe the geometric effect of the transformation produced by the matrix .

• AA dilation with center at the origin and scale factor 3 followed by a reflection in the line
• BA dilation with center at the origin and scale factor 3 followed by a rotation about the origin
• CA dilation with center at the origin and scale factor 3 followed by a reflection in the line
• DA dilation with center at the origin and scale factor 3 followed by a rotation about the origin
• EA dilation with center at the origin and scale factor 3 followed by a rotation about the origin

Q6:

Which of the following compositions of transformations is represented by the matrix ?

• AA rotation of about the origin followed by a reflection in the line
• BA rotation of about the origin followed by a reflection in the line
• CA dilation with a center at the origin and scale factor 2 followed by a reflection in the line
• DA dilation with a center at the origin and scale factor followed by a reflection in the line
• EA dilation with a center at the origin and scale factor followed by a reflection in the line

Q7:

A dilation with center the origin is composed with a rotation about the origin to form a new linear transformation. The transformation formed sends the vector to .

Find the matrix representation of the transformation formed.

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Find the scale factor of the original dilation.

• Ascale factor = 169
• Bscale factor = 13
• Cscale factor = 154
• Dscale factor = 13
• Escale factor =

Q8:

The unit square, with vertices , , , and , is transformed by a rotation and then a dilation. Its image under this combined transformation is , as shown in the diagram. What are the coordinates of ?

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What is the matrix of the combined transformation?

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